VI.1 Pythagoras

b. Samos, Ionia (now Samos, Greece)?, ca. 569 B.C.E.;

d. Metapontum, Magna Graecia (now Metaponto, Italy)?, ca. 494 B.C.E. Incommensurability theorem of Pythagoras

One of the most elusive figures of antiquity, Pythagoras is famous not just for his alleged mathematical achievements: it has been claimed that he had a golden thigh and that he issued a prescription against broad beans. Few things about him can be taken as historical facts, but we can be reasonably confident that he lived in around the sixth century B.C.E. in Greek southern Italy and that he established a group of followers, the Pythagoreans, who shared not just beliefs, but also dietary habits and a code of behavior. The existence of anecdotes about splinter Pythagoreans who revealed secrets to outsiders and were accordingly punished suggests that they were far from constituting a completely homogeneous group.

After a peak period in the late fifth century B.C.E., the Pythagoreans dispersed, probably as a result of their involvement in the public life of various city-states. The impact of their theories about the universe and the soul was very long-lived, though, and can be felt in Plato, Aristotle, and later authors. From the third century B.C.E. until well into late antiquity, a stream of texts was produced that purported to be by Pythagoras or his immediate successors. Indeed, historians talk of a neo-Pythagorean philosophical movement, sometimes associated with neo-Platonism.

The name of Pythagoras and his school is most commonly linked to the theorem establishing that the square on the hypotenuse of a right-angled triangle is equivalent to the sum of the squares on the other two sides. In fact, there is some evidence that the mathematical property expressed by the theorem was known in Mesopotamia long before Pythagoras’s time; the ancient sources attributing the result to him are late and not entirely reliable, and no actual proof of the theorem is found before Euclid’s Elements. While the proof itself may predate EUCLID [VI.2], there is no solid reason to connect it to Pythagoras.

Similarly, the discovery of the incommensurability of the side and the diagonal of a square, often attributed to Pythagoreans, may have been made earlier in Mesopotamia, and the earliest full proof in a Greek context belongs to a later period.

Pythagoras’s real contribution to mathematics lies elsewhere. The Pythagoreans are credited by Aristotle with the theory that “things themselves are numbers.” One interpretation is that they believed that mathematics offered a key to understanding reality, whether this reality was conceived to have an underlying geometrical structure (as in Plato’s Timaeus), or whether it was simply seen as ordered and “in proportion.” Indeed, Pythagoreans are plausibly credited with a strong interest in formulating the numerical ratios of musical concords and harmony. They connected the harmonious sound produced by, say, the plucking of a string with the fact that the musician plucked it at specific, mathematically expressible points. Breaking the mathematical proportion between the points on the string unsettled the sound produced. The heavenly bodies themselves, according to the Pythagoreans, produced music, thanks to their mathematical, and therefore orderly, arrangement. Understand the mathematics, and you will grasp the structure of reality: this insight is perhaps Pythagoras’s true legacy.

Further Reading

Burkert, W. 1972. Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard University Press. (Revised English translation of 1962 Weisheit und Wissenschaft: Studien zu Pythagoras, Philolaos und Platon. Nürnberg: H. Carl.)

Zhmud, L. 1997. Wissenschaft, Philosophie und Religion im frühen Pythagoreismus. Berlin: Akademie.

Serafina Cuomo